If you are getting the error "Errors in the back-end database access module. The size specified for a binding was too small, resulting in one or more column values being truncated. Errors in the OLAP storage engine.” While processing a Cube (SSAS). This is because the size of the Attribute exceeds the Data Size defined in the attribute.
This error can be resolved by increase the value of the property called DataSize for the attribute under both NameColumn and KeyColumns. This can be achieved in 2 different ways.
1. Find the Dimension which causes the error (This can be achieved by examining the error message). Select the Attribute and in the property window change the Data Size of both NameColumn and KeyColumn.
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2. Open SSMS, Connect Analysis service. Expand corresponding Database, then the cube folder and finally the Measure Group. Find the measure which causes error and open it in the XMLA (Shown in below image). Find the field name in which you are getting the error and change the <DataSize> accordingly. Repeat the same if necessary. And execute the XMLA Query.
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Please note, BIDS Helper has a Dimension Data Type Discrepancy Check feature, it may fix this for you:
Dimension Data Type Discrepancy Check